A triangle has two of its sides along the axes, its third side touches the circle x2 + y2 – 2ax – 2ay + a2 = 0. If he locus of the circumcentre of the triangle passes through the point (38, –37) then a2 – 2a is equal to
The given circle has its centre at C (a, a) and its radius is a so that it touches both the axes along which lie the two sides of the triangle. Let third side be
So that A is (p, 0) and B is (0, q) (fig) and the line AB touches the given circle. Since ∠AOB is a right angle, AB is a diameter of the circumcircle of the triangle AOB. So the circumcentre P(h, k) of the triangle AOB is the mid-point of AB.
i.e. 2h = p, 2k = q
Hence the locus of
P(h, k) is a2 – 2a (x + y) + 2xy = 0.
Since it passes through (38, –37)
a2 – 2a = 2 × 38 × 37 = 2812.
If two distinct chords, drawn from the point (p, q) on the circle x2 + y2= px + qy (where pq ≠ 0) are bisected by the x-axis, then
Let A0 A1 A3 A4 A5 be a regular hexagon inscribed in a unit circle with centre at the origin. Then the product of the lengths of the line segmentsA0 A1, A0 A2 and A0 A4 is
C1 and C2 are circles of unit radius with centres at (0, 0) and (1, 0) respectively. C3 is a circle of unit radius, passes through the centres of the circles C1 and C2 and have its centre above x-axis. Equation of the common tangent to C1 and C3 which does not pass through C2 is
A chord of the circle x2 + y2 – 4x – 6y = 0 passing through the origin subtends an angle tan-1 (7/4) at the point where the circle meets positivey-axis.
Equation of the chord is
On the line joining the points A (0, 4) and B (3, 0), a square ABCD is constructed on the side of the line away from the origin. Equation of the circle having centre at C and touching the axis of x is
A circle with centre at the origin and radius equal to a meets the axis of x at A and B.P (α) and Q (β) are two points on this circle so that α – β = 2γ, where γ is a constant. The locus of the point of intersection of APand BQ is
A circle C1 of radius b touches the circle x2 + y2 = a2 externally and has its centre on the positive x-axis; another circle C2 of radius c touches the circle C1 externally and has its centre on the positive x-axis. Given a< b < c, then the three circles have a common tangent if a, b, c are in
Angle of intersection of these circle is
If C1, C2 are the centre of these circles than area of Δ OC1 C2, where Ois the origin, is
Two circles are inscribed and circumscribes about a square ABCD, length of each side of the square is 32. P and Q are points respectively of these circles, then is equal to